By default XCOM: Enemy Unknown is running in a story mode, where you doing the main missions is driving the story and the alien invasion forwards. That has two major disadvantages: The story is always the same. And the story isn't actually that great, especially the end being weak. So some people would prefer to run XCOM in endless mode, just fighting back alien attack after alien attack.

Fortunately doing so is surprisingly easy: Story progress is scripted and depends on your actions, not just the timeline. I checked by simply not doing the alien base assault. You probably will get the possibility to assault the alien base relatively early, before you have seen the stronger aliens. So with time at first you will notice that you encounter stronger and stronger aliens in the attacks. But the good news is that this levels off: If you don't take the alien base, you will never meet the kind of aliens with psi powers, like ethereals.

I've been running XCOM for over a game year without assaulting the alien base, and there is simply nothing bad that happens. I can just endless shoot down one UFO after another or do other missions endlessly. I meet the strongest kind of mutons and nasty enemies like sectopods, but I never get mind-controlled. Of course by now I have tons of all sorts of materials, and the very best armor and weapons available. And I can train lots of soldiers to the highest ranks.

Of course I could also do a similar endless mode, just with psi powers on both sides. I would just have to assault the alien base, and then stop the story at a later point. There appear to be several points in the game where you can reasonably switch from story mode to endless mode with no ill effect. So if you just want to do tactical battles without finishing the story, the choice is yours.